#219908

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1174 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)(7,13,14,10)(8,16)(9,12)(11,17)(15,18,20,19)(21,22)(23,29,30,36)(24,26)(25,28)(27,31,34,33)(32,35);(1,2)(3,4)(7,14)(10,13)(15,20)(18,19)(22,26)(27,34)(28,35)(29,36);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,19)(17,20)(21,23)(22,27)(24,30)(25,33)(26,34)(28,36)(29,35)(31,32);(2,4)(5,8,9,17)(6,11,12,16)(7,15,14,20)(10,18,13,19)(22,26)(27,34)(28,35)(29,36)

Polycyclic presentation:

2^9|a^2=g,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, 127104);