#375155

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1820 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=fh,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264920, #72264940, #72264987
maximal normal subgroup of	#72264920, #72264940, #72264987

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282351);