#6375533

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
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	C₂×#3002	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2878 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)(7,11)(8,10)(9,12)(14,19)(15,23)(16,18)(20,27)(24,28)(25,26);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,14)(15,24)(16,21)(17,19)(18,22)(20,25)(23,27)(26,28);(7,12)(9,11)(13,15)(14,20)(16,23)(17,25)(18,26)(19,24)(21,28)(22,27);(6,10)(9,11)(13,16,17,18)(14,21,19,22)(15,23,25,26)(20,28,24,27);(13,16,17,18)(14,22,19,21)(15,23,25,26)(20,27,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6282729);