#7623905

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#33579	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322566
maximal normal subgroup of	#72322566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531101);