#233758

order	512
nilpotency class	3
isoclinism	isoclinic class of #233754
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	#36684	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1302 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1302 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)(15,23,26,37)(16,20,27,35)(17,29,32,21)(19,30,34,25)(24,40,41,39)(28,38,31,36)(33,43,44,42);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,15)(14,19)(16,24)(17,30)(18,26)(20,33)(21,37)(22,34)(23,29)(25,32)(27,41)(28,43)(31,42)(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)(25,43)(26,41)(28,32)(29,36)(30,42)(34,44)(37,40);(6,11)(10,12)(16,27)(20,35)(24,41)(28,31)(33,44)(36,38)(39,40)(42,43)

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=dhi,e^b=eh,f^b=fi,d^c=dh,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, 140954);