#365321

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

988 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)(7,17,20,28)(8,21)(9,12)(10,15)(11,16)(13,19,29,25)(14,22)(18,27,33,36)(23,30)(24,26)(31,35,34,32);(1,2)(3,4)(6,12,16,15)(13,17,29,28)(14,26,30,21)(27,31,36,34);(2,4)(5,7)(6,13)(8,18)(9,19)(10,25)(11,20)(12,17)(14,27)(15,28)(16,29)(21,34)(22,32)(23,35)(24,33)(26,31)(30,36);(5,8,9,22,11,24,10,23)(6,14,15,21,16,30,12,26)(7,18,19,32,20,33,25,35)(13,27,28,34,29,36,17,31)

Polycyclic presentation:

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

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, 272517);