#378100

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ei,d^2=fhi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fi,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285296);