#711884

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4622 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)(10,11)(14,15)(20,24)(22,23);(2,4)(6,8)(9,10,13,11)(12,15,16,14)(17,18,21,19)(20,22,24,23);(1,2)(3,4)(5,6)(7,8)(10,11)(14,15)(17,19,21,18)(20,23,24,22);(9,11,13,10)(12,15,16,14)(17,20)(18,22)(19,23)(21,24);(9,12)(10,14)(11,15)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280318, #72280339, #72280430, …
maximal normal subgroup of	#72280318, #72280339, #72280430, …
maximal quotient of	C₂×#711884

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 619080);