#245137

order	512
nilpotency class	3
isoclinism	isoclinic class of #245126
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
prime radical	#36706	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1389 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)(3,6)(4,7)(5,8)(9,15)(10,12)(11,14)(13,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,32)(29,31);(4,11)(6,12)(7,14)(9,16)(22,31)(25,32);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)(17,19,21,26)(18,22,24,31)(20,29,30,27)(23,32,28,25);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,16)(12,15)(20,30)(23,28)(25,32)(27,29)

Polycyclic presentation:

2^9|c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=dhi,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152333);