#366326

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₄×#120
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1866 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265308, #72307320, #72307343
maximal normal subgroup of	#72265308, #72307320, #72307343

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273522);