#369953

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
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×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(9,10,16,19,15,20,14,11)(12,21,26,17,25,23,30,33)(13,18,28,27,29,34,31,22)(24,36,39,40,37,35,38,32);(1,2,4,6)(3,7,8,5)(9,11,14,20,15,19,16,10)(12,23,30,17,25,21,26,33)(13,27,31,18,29,22,28,34)(24,35,38,40,37,36,39,32);(1,3,4,8)(2,5,6,7)(9,12,15,25)(10,17,20,33)(11,21,19,23)(13,24,29,37)(14,30,16,26)(18,32,34,40)(22,35,27,36)(28,39,31,38);(9,13)(10,18)(11,22)(12,24)(14,31)(15,29)(16,28)(17,32)(19,27)(20,34)(21,35)(23,36)(25,37)(26,39)(30,38)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,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	#72265390, #72307320, #72307343
maximal normal subgroup of	#72265390, #72307320, #72307343

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277149);