#6400067

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	#2408	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2059 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)(5,6,9,12)(7,11,8,10)(13,14,19,17,18,22,16,21)(15,23,24,26,27,20,25,28);(5,6)(7,10)(8,11)(9,12)(13,15)(14,20)(16,25)(17,28)(18,27)(19,24)(21,26)(22,23);(2,4)(5,7,9,8)(6,10,12,11)(13,16,18,19)(14,17,22,21)(15,24,27,25)(20,26,23,28);(5,7,9,8)(6,11,12,10)(13,16,18,19)(14,21,22,17)(15,25,27,24)(20,26,23,28);(5,8,9,7)(6,11,12,10)(13,17,19,22,18,21,16,14)(15,26,24,20,27,28,25,23)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,e^b=ei,e^c=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, 6307263);