#245581

order	512
nilpotency class	3
isoclinism	isoclinic class of #245567
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	#43491	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1237 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,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,e^c=ei,e^d=eh,f^d=fi

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, 152777);