#248626

order	512
nilpotency class	3
isoclinism	isoclinic class of #248599
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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

1313 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,d^c=di,e^d=eh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306408, #72306411, #72306487
maximal normal subgroup of	#72306408, #72306411, #72306487

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155822);