#230180

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=hi,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,e^b=eh,g^b=gh,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137376);