#609828

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
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	C₂×#2397	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3454 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=h,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280075, #72280572, #72280933, …
maximal normal subgroup of	#72280075, #72280572, #72280933, …
maximal quotient of	C₂×#609828

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 517024);