#6371972

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342612
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3497	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

5049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,11,17,24)(8,12)(9,13)(10,14)(15,22,30,34)(16,23,32,36)(18,25)(19,26)(20,27)(21,28)(29,33,31,35);(5,7)(6,11,14,24)(8,15)(9,16)(10,17)(12,22,26,34)(13,23,28,36)(18,29)(19,30)(20,31)(21,32)(25,33,27,35);(1,2,3,4);(5,8,10,19)(6,12,14,26)(7,15,17,30)(9,20,21,18)(11,22,24,34)(13,27,28,25)(16,31,32,29)(23,35,36,33);(5,9,10,21)(6,13,14,28)(7,16,17,32)(8,18,19,20)(11,23,24,36)(12,25,26,27)(15,29,30,31)(22,33,34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,d^2=i,e^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=hi,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, 6279168);