#361711

order	512
nilpotency class	3
isoclinism	isoclinic class of #188252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3566 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^c=hi,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, 268907);