#219917

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1077 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)(8,15,17,12)(9,18)(10,13)(14,20,16,19)(21,22)(23,30)(24,29)(25,27)(26,33)(28,32)(31,36)(34,35);(1,2)(3,4)(5,7,10,9)(6,11,13,18)(8,16,17,14)(12,20,15,19)(22,27)(23,31)(29,33)(32,34);(5,8,10,17)(6,12,13,15)(7,14,9,16)(11,19,18,20)(21,23,25,31)(22,28,27,35)(24,32,26,34)(29,36,33,30);(2,4)(5,9,10,7)(6,11,13,18)(8,16,17,14)(12,19,15,20)(21,24,25,26)(22,29,27,33)(23,32,31,34)(28,36,35,30)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=ch,d^b=dei,f^c=fi,g^c=ghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127113);