#393419

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

954 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,c^b=ci

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, 300615);