#7626303

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
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	C₂×#3500	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8358 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533499);