#277400

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

605 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ehi,c^2=i,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 184596);