#374569

order	512
nilpotency class	3
isoclinism	isoclinic class of #195990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=efh,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,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, 281765);