#185979

order	512
nilpotency class	3
isoclinism	isoclinic class of #18132
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2354 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93175);