#6597030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574031
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36096	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2496 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)(7,10)(8,11)(9,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,23)(24,27)(25,28);(5,7)(6,9)(8,12)(10,11)(13,15,17,23,18,24,16,25)(14,19,20,22,21,28,26,27);(2,4)(7,12)(9,10);(5,7)(6,10)(8,12)(9,11)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,28,22);(7,12)(9,10)

Polycyclic presentation:

2^9|b^2=fh,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257226, #72257264, #72263867, …
maximal normal subgroup of	#72257226, #72257264, #72263867, …
maximal quotient of	#72264289

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504226);