#6597025

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2360 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307406, #72307677, #72307778
maximal normal subgroup of	#72307406, #72307677, #72307778

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504221);