#6372409

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279605);