#6398665

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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×#546	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2079 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)(13,14)(15,23)(16,21)(17,22)(18,27)(19,25)(20,24)(26,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,26)(17,25)(18,24)(20,21)(22,23)(27,28);(13,15)(14,20)(16,24)(17,25)(18,26)(19,21)(22,28)(23,27);(7,12)(9,11)(13,16,17,18)(14,21,22,27)(15,24,25,26)(19,28,23,20);(1,3,4,2)(6,10)(9,11)(13,16,17,18)(14,21,22,27)(15,24,25,26)(19,28,23,20)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6305861);