#298369

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

988 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205565);