#376011

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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1250 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265222, #72307302, #72307343
maximal normal subgroup of	#72265222, #72307302, #72307343

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283207);