#361783

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6)(7,11)(8,10)(9,12)(13,14,17,22,18,20,16,21)(15,19,23,28,25,27,24,26);(1,2,3,4)(5,7,8,12)(6,9,10,11)(13,15,16,24,18,25,17,23)(14,19,21,26,20,27,22,28);(2,4)(14,20)(15,23)(16,17)(19,26)(24,25)(27,28);(6,10)(9,11)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238763, #72239135, #72239138
maximal normal subgroup of	#72238763, #72239135, #72239138

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268979);