#272968

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1017 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fh,c^2=hi,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180164);