#7330689

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(9,10)(11,13)(12,14)(15,16)(18,23)(19,26)(21,32)(24,28)(27,31)(29,30);(1,2)(3,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(17,18)(19,27)(20,21)(22,23)(24,30)(25,32)(26,28)(29,31);(1,3)(2,5)(4,8)(6,7)(10,11)(12,16)(17,19,21,26,22,30,32,29)(18,24,20,31,23,27,25,28);(2,6)(3,8)(9,12,13,16)(10,14,11,15)(17,20,22,25)(18,21,23,32)(19,28,30,31)(24,29,27,26);(2,6)(5,7)(10,11)(14,15)(17,21,22,32)(18,25,23,20)(19,29,30,26)(24,31,27,28)

Polycyclic presentation:

2^9|b^2=fi,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308644, #72308985
maximal normal subgroup of	#72308644, #72308985

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7237885);