#158319

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

5456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

(6,8)(9,10)(11,12)(17,18,21,24)(19,23,20,22);(2,4)(6,8)(10,12)(14,16)(17,18)(19,23)(20,22)(21,24);(13,14)(15,16)(17,19,21,20)(18,22,24,23);(1,2)(3,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16)(17,20,21,19)(18,23,24,22)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323679, #72323681, #72323683, …
maximal normal subgroup of	#72323679, #72323681, #72323683, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 65515);