#778810

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4307 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331172, #72331191, #72331248
maximal normal subgroup of	#72331172, #72331191, #72331248
maximal quotient of	#72332998

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 686006);