#608441

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15172	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304057, #72304094, #72304112
maximal normal subgroup of	#72304057, #72304094, #72304112

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515637);