#154849

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

2710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=f,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278999, #72279050, #72279137
maximal normal subgroup of	#72278999, #72279050, #72279137
maximal quotient of	#72299914

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 62045);