#298172

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

966 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)(5,6,9,13)(7,12,15,11)(8,17,16,20)(10,19,18,14)(21,22)(23,29)(24,27)(25,34)(26,31)(28,33)(30,35)(32,36);(1,2,4,3)(5,7)(6,11)(8,10)(9,15)(12,13)(14,17)(16,18)(19,20)(21,23)(22,26)(24,31)(25,35)(27,29)(28,30)(32,33)(34,36);(5,8,18,15,9,16,10,7)(6,12,19,20,13,11,14,17)(21,22)(23,30,31,36)(24,27)(25,28)(26,35,29,32)(33,34);(1,3,4,2)(6,13)(7,15)(8,16)(14,19)(22,27)(26,29)(28,34)(30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,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,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205368);