#7626924

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	C₂×#3518	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=gi,f^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, 7534120);