#213549

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1110 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)(3,7)(4,6)(5,8)(9,10,12,14,11,13,15,16)(17,18,21,25,22,26,20,29)(19,23,27,32,28,24,30,31);(1,3)(2,5)(4,8)(6,7)(10,13)(14,16)(17,19,22,28)(18,23,26,24)(20,30,21,27)(25,32,29,31);(3,8)(5,7)(17,19)(18,24)(20,27)(21,30)(22,28)(23,26)(25,32)(29,31);(2,6)(5,7)(17,20,22,21)(18,25,26,29)(19,27,28,30)(23,31,24,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^b=dh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120745);