#361720

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#577	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=i,e^2=i,g^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,g^c=gi,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, 268916);