#7597025

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

2314 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)(16,21)(19,25)(20,26)(23,28)(24,27);(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,16,18,17,20,21,26)(15,19,24,25,22,23,27,28);(1,2,4,3)(6,9)(10,11)(13,15)(14,19)(16,24)(17,22)(18,25)(20,23)(21,27)(26,28);(1,3,4,2)(5,7,8,12)(6,10,9,11)(14,20)(15,22)(18,26)(24,27);(13,14,21,26,17,20,16,18)(15,23,27,25,22,19,24,28)

Polycyclic presentation:

2^9|b^2=f,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,e^b=ei,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	#72340631, #72340639, #72340657
maximal normal subgroup of	#72340631, #72340639, #72340657

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504221);