#392613

order	512
nilpotency class	3
isoclinism	isoclinic class of #32894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3458 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299809);