#7518956

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 16 points:

(5,6)(7,8)(9,10,15,16)(11,12,13,14);(1,2)(3,4)(5,6)(7,8)(9,11,14,10,15,13,12,16);(6,8)(10,16)(11,13);(2,4)(6,8)(9,12,15,14)(10,11,16,13);(9,13,14,16,15,11,12,10)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340637, #72340648, #72340678
maximal normal subgroup of	#72340637, #72340648, #72340678

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426152);