#7626582

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

6996 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)(15,22)(19,26)(23,27)(24,28);(6,7)(10,11)(13,14,18,21)(15,19,22,26)(16,17,20,25)(23,24,27,28);(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,23)(17,24)(18,22)(20,27)(21,26)(25,28);(5,8,9,12)(6,10,7,11)(13,16)(14,17)(15,23)(18,20)(19,24)(21,25)(22,27)(26,28);(13,17)(14,20)(15,24)(16,21)(18,25)(19,27)(22,28)(23,26);(13,15)(14,19)(16,23)(17,24)(18,22)(20,27)(21,26)(25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=g,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, 7533778);