#298078

order	512
nilpotency class	3
isoclinism	isoclinic class of #271259
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×#127	D₄
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)(7,13,15,20)(9,17,14,12)(10,18,19,16)(21,22,24,27)(23,26,31,35)(25,33,32,28)(29,34,30,36);(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,19)(18,20)(21,23)(22,26)(24,31)(25,29)(27,35)(28,36)(30,32)(33,34);(5,7,14,18,8,15,9,16)(6,10,17,20,11,19,12,13)(22,27)(23,29,31,30)(26,34,35,36)(28,33);(1,3,4,2)(6,11)(7,15)(12,17)(16,18)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=fhi,d^2=eh,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=dh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205274);