#230001

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137197);