#188474

order	512
nilpotency class	3
isoclinism	isoclinic class of #161128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1108 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95670);