#358740

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265936);