#7564641

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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	#25397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7471837);