#648518

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3644 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=f,d^2=g,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	#72306173, #72306249, #72306258
maximal normal subgroup of	#72306173, #72306249, #72306258

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 555714);