#382377

order	512
nilpotency class	3
isoclinism	isoclinic class of #29610
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3646 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289573);