#7626625

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533821);