#6371598

order	512
nilpotency class	3
isoclinism	isoclinic class of #60367
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×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	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2522 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=di,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278794);