#361693

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3632 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)(7,16)(8,20,23,31)(9,24)(10,14)(11,15)(12,18)(13,22,32,26)(17,30,21,27)(19,28)(25,29)(33,35,36,34);(1,2,3,4)(5,7,11,19)(6,12,15,29)(8,21,23,17)(9,25,10,18)(13,30,32,27)(14,28,24,16)(20,36,31,33)(22,34,26,35);(2,4)(5,8)(6,13)(7,17)(9,22)(10,26)(11,23)(12,27)(14,31)(15,32)(16,33)(18,34)(19,21)(20,24)(25,35)(28,36)(29,30);(5,7)(6,12)(8,17)(9,25)(10,18)(11,19)(13,27)(14,28)(15,29)(16,24)(20,33)(21,23)(22,35)(26,34)(30,32)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fi,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, 268889);