#6382052

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1977 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)(6,10)(11,12)(13,14,19,22,18,21,28,26)(15,17,20,27,23,16,24,25);(5,6)(7,11)(8,12)(9,10)(13,15,19,20,18,23,28,24)(14,16,22,25,21,17,26,27);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,17)(14,20,21,24)(15,22,23,26)(19,25,28,27);(5,8,9,7)(6,11,10,12)(13,17,18,16)(14,20,21,24)(15,22,23,26)(19,27,28,25);(14,21)(16,17)(22,26)(25,27)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,e^c=ei,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289248);