#378877

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14,20,26,18,24,19,25)(15,27,30,23,29,21,17,35)(16,22,34,41,33,40,36,42)(28,43,31,39,44,37,32,38);(6,10)(11,12)(13,15,19,17,18,29,20,30)(14,21,25,23,24,27,26,35)(16,31,34,44,33,32,36,28)(22,38,41,43,40,39,42,37);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,33)(14,22,24,40)(15,28,29,44)(17,32,30,31)(19,36,20,34)(21,37,27,43)(23,39,35,38)(25,42,26,41);(5,8,9,7)(6,11,10,12)(13,17,19,29,18,30,20,15)(14,23,25,27,24,35,26,21)(16,32,36,44,33,31,34,28)(22,39,42,43,40,38,41,37)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216185
maximal normal subgroup of	#72216185
maximal quotient of	#72216074

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286073);