#7596908

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2160 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281862, #72282293, #72308936
maximal normal subgroup of	#72281862, #72282293, #72308936

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504104);