#408196

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=efh,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307246, #72307320, #72307328
maximal normal subgroup of	#72307246, #72307320, #72307328

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315392);