#7149193

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2116 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)(7,11)(10,12)(13,14,17,21,16,18,24,19)(15,20,23,27,22,25,28,26);(1,2,3,4)(6,9)(7,11)(14,18)(19,21)(20,25)(26,27);(5,6)(7,10)(8,9)(11,12)(14,19)(17,24)(18,21)(20,26)(23,28)(25,27);(2,4)(6,9)(10,12)(13,15,16,22)(14,20,18,25)(17,23,24,28)(19,26,21,27);(5,7,8,11)(6,10,9,12)(14,18)(15,22)(19,21)(23,28)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307631, #72307710, #72308233
maximal normal subgroup of	#72307631, #72307710, #72308233

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056389);