#245278

order	512
nilpotency class	3
isoclinism	isoclinic class of #245246
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

1244 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1244 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,22,26)(19,23,29,36)(20,24,31,39)(21,32,33,25)(27,34,41,37)(28,42,43,35)(30,38,44,45)(40,46,48,47);(1,3,5,10)(2,6,8,12)(4,9,11,15)(7,13,14,16)(17,19,22,29)(18,23,26,36)(20,27,31,41)(21,28,33,43)(24,37,39,34)(25,35,32,42)(30,40,44,48)(38,47,45,46);(6,12)(13,16)(17,20)(18,24)(19,27)(21,30)(22,31)(23,34)(25,38)(26,39)(28,40)(29,41)(32,45)(33,44)(35,46)(36,37)(42,47)(43,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21)(18,25)(19,28)(20,30)(22,33)(23,35)(24,38)(26,32)(27,40)(29,43)(31,44)(34,46)(36,42)(37,47)(39,45)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268306
maximal normal subgroup of	#72268306
maximal quotient of	#72268310

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152474);