#374839

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

984 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei

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, 282035);