#245599

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	#3505	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1166 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)(3,11)(4,7)(5,15)(6,10)(8,13)(9,12)(14,16)(17,18)(19,22)(20,23)(21,24)(25,31)(26,28)(27,30)(29,32);(1,3,6,13)(2,7,10,16)(4,5,14,12)(8,15,11,9)(22,28)(29,31);(17,19,21,26)(18,22,24,28)(20,25,27,32)(23,29,30,31);(1,4,6,14)(2,8,10,11)(3,12,13,5)(7,9,16,15)(17,20)(18,23)(19,25)(21,27)(22,29)(24,30)(26,32)(28,31)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,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, 152795);