#359684

order	512
nilpotency class	3
isoclinism	isoclinic class of #23892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3586	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3687 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)(6,12)(8,21)(9,10)(13,26)(15,16)(17,28)(18,32)(19,24)(22,23)(25,29)(27,36)(33,34);(5,6)(7,17)(8,14)(9,16)(10,12)(11,15)(13,19)(18,31)(20,26)(21,25)(22,30)(23,29)(24,28)(27,33)(32,36)(34,35);(5,7)(6,13)(8,18)(9,19)(10,24)(11,20)(12,17)(14,27)(15,28)(16,26)(21,32)(22,34)(23,33)(25,31)(29,35)(30,36);(5,8,11,22)(6,14,15,30)(7,18,20,34)(9,23,10,21)(12,25,16,29)(13,27,28,36)(17,31,26,35)(19,33,24,32)

Polycyclic presentation:

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

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, 266880);