#1494270

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	C₂×#2668	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3554 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)(9,10)(11,16)(12,13)(14,15)(17,18,20,22)(19,23,24,21);(6,8)(10,15)(13,16)(17,18)(19,23)(20,22)(21,24);(5,6)(7,8)(9,11,14,12)(10,16,15,13)(17,18)(19,21)(20,22)(23,24);(1,2,3,4)(9,12,14,11)(10,16,15,13)(17,18)(19,23)(20,22)(21,24);(9,13,14,16)(10,12,15,11)(17,19)(18,21)(20,24)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304062, #72304089, #72304113
maximal normal subgroup of	#72304062, #72304089, #72304113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1401466);