#245132

order	512
nilpotency class	3
isoclinism	isoclinic class of #245126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1233 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1233 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152328);