#6380525

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2166 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)(5,6)(7,10)(8,12)(9,11)(13,14)(15,20)(16,21)(17,19)(18,22)(23,27)(24,28)(25,26);(5,7,9,8)(6,10,11,12)(14,19)(16,22)(17,21)(20,26)(23,28)(25,27);(5,8,9,7)(6,10,11,12)(13,15)(14,20)(16,23)(17,25)(18,24)(19,26)(21,27)(22,28);(13,16,18,19)(14,21,22,17)(15,23,24,26)(20,27,28,25);(15,24)(20,28)(23,26)(25,27)

Polycyclic presentation:

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

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, 6287721);