#6629729

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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	C₂×#3050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2548 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:

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

Polycyclic presentation:

2^9|b^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240658, #72240668, #72240672, …
maximal normal subgroup of	#72240658, #72240668, #72240672, …
maximal quotient of	#72308727, #72309080

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536925);