#6687696

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682564
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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2078 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)(7,12)(8,11)(9,10)(13,14,18,22)(15,23,25,19)(16,20,17,21)(24,28,26,27);(1,2)(3,4)(6,10)(7,11)(13,15)(14,19)(16,24)(17,26)(18,25)(20,27)(21,28)(22,23);(2,4)(5,7,12,6,9,11,8,10)(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,27,23,28);(6,10)(7,11)(13,16,18,17)(14,21,22,20)(15,24,25,26)(19,28,23,27);(2,4)(5,8,9,12)(6,11,10,7)(13,17,18,16)(14,21,22,20)(15,24,25,26)(19,27,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307339, #72307379, #72308015
maximal normal subgroup of	#72307339, #72307379, #72308015

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594892);