#7501068

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2907 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^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,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307394, #72307498, #72307595
maximal normal subgroup of	#72307394, #72307498, #72307595

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408264);