#6446617

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#532	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305369, #72305373, #72305381
maximal normal subgroup of	#72305369, #72305373, #72305381
maximal quotient of	#72305816

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353813);