#245682

order	512
nilpotency class	3
isoclinism	isoclinic class of #245615
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fh,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312035, #72312041, #72312060
maximal normal subgroup of	#72312035, #72312041, #72312060

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152878);