#6379602

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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×#3517	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286798);