#6393504

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
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×#3548	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3362 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)(7,11)(8,10)(9,12)(13,14)(15,16)(17,21)(18,22)(19,28)(20,23)(24,26)(25,27);(5,7)(6,9)(8,12)(10,11)(13,15)(14,16)(17,18)(19,24)(20,25)(21,22)(23,27)(26,28);(2,4)(13,16,19,21,20,27,18,26)(14,17,22,15,23,24,28,25);(7,12)(9,11)(13,17,20,24)(14,21,23,26)(15,18,25,19)(16,22,27,28);(6,10)(9,11)(13,18,20,19)(14,22,23,28)(15,24,25,17)(16,26,27,21)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6300700);