#7596806

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2221 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314242
maximal normal subgroup of	#72314242

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504002);