#188516

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1060 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)(8,16)(9,10)(12,17)(14,18,20,19)(21,22)(23,32)(24,31)(25,30)(26,29)(27,34)(28,33)(35,36);(1,2,3,4)(6,10)(11,13)(12,16)(18,19)(21,23)(22,27)(24,35)(25,34)(26,33)(28,31)(29,36)(30,32);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,23)(22,28)(24,33)(25,34)(26,35)(27,31)(29,32)(30,36);(5,8,9,17)(6,12,10,16)(7,14,15,20)(11,18,13,19)(21,24,25,26)(22,29,30,31)(23,33,34,35)(27,28,32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=di,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, 95712);