#378488

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

891 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=efh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,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, 285684);