#6445045

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

2192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307267, #72307285, #72307652
maximal normal subgroup of	#72307267, #72307285, #72307652

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352241);