#6445176

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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

2210 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)(14,19)(15,22)(18,21)(20,23)(24,28)(25,27);(1,2)(3,4)(6,9)(10,12)(13,14)(15,20)(16,18)(17,19)(21,26)(22,25)(23,28)(24,27);(2,4)(13,15,18,22,17,24,21,28)(14,20,16,25,19,27,26,23);(2,4)(13,16)(14,21)(15,23)(17,26)(18,19)(20,22)(24,25)(27,28);(5,7,8,11)(6,10,9,12)(14,19)(15,24)(16,26)(22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307108, #72307249, #72307315
maximal normal subgroup of	#72307108, #72307249, #72307315

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352372);