#353025

order	512
nilpotency class	3
isoclinism	isoclinic class of #352081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dhi,d^c=dh

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, 260221);