#7522982

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1898 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7430178);