#664641

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#546	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|c^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261360, #72261496, #72261775, …
maximal normal subgroup of	#72261360, #72261496, #72261775, …
maximal quotient of	#72333138

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 571837);