#204247

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1368 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)(7,10,14,17)(8,11)(9,12)(13,16,19,20)(15,18)(21,22,25,28,24,27,29,35)(23,26,31,34,30,33,32,36);(5,7,9,14)(6,10)(8,13,15,19)(11,16)(12,17)(18,20)(21,23,29,31,24,30,25,32)(22,26,28,36,27,33,35,34);(1,2,3,4)(5,8,9,15)(6,11,12,18)(7,13,14,19)(10,16,17,20)(22,27)(26,33)(28,35)(34,36);(8,15)(11,18)(13,19)(16,20)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=dhi

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, 111443);