#7597003

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

2480 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)(14,18)(16,19)(20,25)(21,26)(23,28)(24,27);(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,19,26,17,21,16,18)(15,20,27,28,22,23,24,25);(1,2,4,3)(6,9)(10,11)(13,15)(14,20)(16,24)(17,22)(18,25)(19,27)(21,23)(26,28);(1,3,4,2)(5,7,8,12)(6,10,9,11)(14,21)(15,22)(18,26)(24,27);(13,14,19,26,17,21,16,18)(15,23,27,25,22,20,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251206, #72251222, #72256885, …
maximal normal subgroup of	#72251206, #72251222, #72256885, …
maximal quotient of	#72264748, C₂×#7597003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504199);