#454655

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1970 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)(5,6,8,9)(7,10,11,12)(13,14,19,20,17,21,18,22)(15,23,24,27,25,16,26,28);(6,9)(10,12)(13,15)(14,16)(17,25)(18,26)(19,24)(20,28)(21,23)(22,27);(2,4)(5,7)(6,10)(8,11)(9,12)(14,20)(15,24)(16,23)(18,19)(21,22)(25,26);(7,11)(10,12)(13,16)(14,15)(17,23)(18,28)(19,27)(20,26)(21,25)(22,24)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361851);