#370544

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1279 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)(8,12)(9,11)(13,14,16,21,18,22,17,20)(15,23,26,24,27,28,25,19);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,16,26,18,27,17,25)(14,19,21,23,22,24,20,28);(2,4)(5,8,9,7)(6,10,11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,26)(24,25)(27,28);(13,14)(15,24)(16,21)(17,20)(18,22)(19,27)(23,25)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277740);