#384843

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292039);