#378890

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1270 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264661, #72264667, #72264669, …
maximal normal subgroup of	#72264661, #72264667, #72264669, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286086);