#346824

order	512
nilpotency class	3
isoclinism	isoclinic class of #346786
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#537	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271575
maximal normal subgroup of	#72271575

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254020);