#7195708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2502 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264307, #72264378, #72264499
maximal normal subgroup of	#72264307, #72264378, #72264499

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7102904);