#6445171

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2258 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)(7,10)(8,9)(11,12)(14,18)(16,20)(19,25)(21,26)(22,28)(24,27);(1,2)(3,4)(6,9)(10,12)(14,19)(18,25)(21,22)(26,28);(2,4)(13,14,20,25,17,19,16,18)(15,22,27,26,23,21,24,28);(2,4)(13,15)(14,21)(16,24)(17,23)(18,26)(19,22)(20,27)(25,28);(5,7,8,11)(6,10,9,12)(14,19)(15,23)(18,25)(24,27)

Polycyclic presentation:

2^9|c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307139, #72307149, #72307652
maximal normal subgroup of	#72307139, #72307149, #72307652

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352367);