#298020

order	512
nilpotency class	3
isoclinism	isoclinic class of #271643
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1043 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=fh,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=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, 205216);