#245302

order	512
nilpotency class	3
isoclinism	isoclinic class of #245282
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1236 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1236 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,6,10,13)(4,11,12,7)(9,14,15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,37)(28,35)(29,36)(30,38)(31,39)(32,40)(33,41)(34,43)(42,47)(44,46)(45,48);(6,13)(14,16)(17,19,22,29)(18,23,26,36)(20,27,31,43)(21,28,33,45)(24,37,39,34)(25,35,41,48)(30,42,32,44)(38,47,40,46);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,20,22,31)(18,24,26,39)(19,27,29,43)(21,32,33,30)(23,34,36,37)(25,40,41,38)(28,44,45,42)(35,47,48,46);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,14,13,16)(17,21,22,33)(18,25,26,41)(19,28,29,45)(20,30,31,32)(23,35,36,48)(24,38,39,40)(27,42,43,44)(34,46,37,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=hi,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268106
maximal normal subgroup of	#72268106
maximal quotient of	#72268111

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152498);