#233809

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37608	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1308 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1308 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,d^c=di,e^c=ei,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141005);