#221415

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=hi,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,g^c=gh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128611);