#245806

order	512
nilpotency class	3
isoclinism	isoclinic class of #245711
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	C₂×C₄²	C₂⁴
prime radical	#45271	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1150 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281514, #72281547, #72281634
maximal normal subgroup of	#72281514, #72281547, #72281634

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153002);