#182977

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

991 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)(3,11)(4,9)(5,10)(6,8)(7,13)(12,15)(14,16)(17,18,21,26,23,28,22,27)(19,29,25,32,31,20,30,24);(1,3,5,14,6,12,4,13)(2,7,9,15,8,16,10,11)(17,19,22,30,23,31,21,25)(18,24,27,20,28,32,26,29);(2,8)(3,12)(9,10)(13,14)(17,20,21,24,23,29,22,32)(18,25,26,31,28,30,27,19);(1,4,6,5)(2,9,8,10)(3,13,12,14)(7,15,16,11)(17,21,23,22)(18,26,28,27)(19,25,31,30)(20,24,29,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=ch,e^a=eh,g^a=ghi,c^b=ch,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90173);