#358739

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
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×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 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)(13,14,18,23,19,24,17,22)(15,20,27,35,26,34,31,39)(16,21,29,37,30,38,32,40)(25,33,42,44,41,43,28,36);(1,2,4,3)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,28)(17,31)(18,27)(19,26)(21,36)(22,39)(23,35)(24,34)(25,29)(30,42)(32,41)(33,37)(38,44)(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)(22,40)(23,37)(24,38)(26,41)(27,42)(28,31)(34,43)(35,44)(36,39);(5,8,9,7)(6,10,12,11)(15,26)(20,34)(25,41)(27,31)(28,42)(33,43)(35,39)(36,44)

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,c^b=cg,d^b=di,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, 265935);