#277684

order	512
nilpotency class	3
isoclinism	isoclinic class of #264427
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

607 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)(9,18)(10,12)(13,16)(14,19)(17,20);(5,7,9,8,10,16,19,17)(6,11,14,15,12,20,18,13)(21,22,25,28)(23,29,31,36)(24,32,34,27)(26,33,35,30);(2,4)(6,12)(7,16)(8,17)(14,18)(21,23,25,31)(22,26,28,35)(24,33,34,30)(27,36,32,29);(5,8,9,16,10,17,19,7)(6,13,14,11,12,15,18,20)

Polycyclic presentation:

2^9|a^2=fh,b^2=ei,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,d^b=dh,g^b=gi,d^c=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 184880);