#253257

order	512
nilpotency class	3
isoclinism	isoclinic class of #253219
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	C₂³×C₄	C₂⁴
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1061 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160453);