#253630

order	512
nilpotency class	3
isoclinism	isoclinic class of #253603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	C₂×#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

735 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160826);