#367418

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

836 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=ef,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274614);