#7501798

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2634 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)(7,9)(8,10)(11,12)(13,14,19,22)(15,23,20,28)(16,24,26,25)(17,21,18,27);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14,19,22)(15,16,20,26)(17,27,18,21)(23,24,28,25);(2,4)(5,8,9,12)(6,10,7,11)(13,15,19,20)(14,16,22,26)(17,25,18,24)(21,28,27,23);(6,7)(10,11)(13,16)(14,20)(15,22)(17,23)(18,28)(19,26)(21,25)(24,27);(13,17,19,18)(14,21,22,27)(15,24,20,25)(16,23,26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315711, #72315744, #72315805
maximal normal subgroup of	#72315711, #72315744, #72315805

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408994);