#6400993

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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	#52160	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1910 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)(7,10)(8,12)(9,11)(13,14,19,23,18,22,16,20)(15,24,21,27,25,17,26,28);(5,7,10,6)(8,12,9,11)(13,15,18,25)(14,17,22,24)(16,26,19,21)(20,27,23,28);(2,4)(5,8,10,9)(6,11,7,12)(13,16,18,19)(14,20,22,23)(15,21,25,26)(17,28,24,27);(5,8,10,9)(6,12,7,11)(13,16,18,19)(14,20,22,23)(15,21,25,26)(17,28,24,27);(5,9,10,8)(6,12,7,11)(13,17,16,27,18,24,19,28)(14,21,20,15,22,26,23,25)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6308189);