#7533710

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2107 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6)(7,11)(8,10)(9,12)(13,14,19,21)(15,22,17,18)(16,24,26,20)(23,27,25,28);(1,2)(3,4)(7,12)(9,11)(13,15,18,14,19,17,22,21)(16,25,27,24,26,23,28,20);(5,7,8,12)(6,9,10,11)(13,16)(14,20)(15,23)(17,25)(18,27)(19,26)(21,24)(22,28);(2,4)(6,10)(9,11)(14,21)(15,17)(16,26)(27,28);(13,17,18,21,19,15,22,14)(16,25,27,24,26,23,28,20)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281938
maximal normal subgroup of	#72281938

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7440906);