#7626671

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×#3505	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7222 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)(6,9)(10,12)(13,14,20,23)(15,24,18,21)(16,19,17,22)(25,26,27,28);(1,3,4,2)(13,15,20,18)(14,21,23,24)(16,27,17,25)(19,26,22,28);(5,6)(7,10)(8,9)(11,12)(13,16,20,17)(14,22,23,19)(15,25,18,27)(21,26,24,28);(7,11)(10,12)(13,17,20,16)(14,19,23,22)(15,25,18,27)(21,26,24,28);(5,7)(6,10)(8,11)(9,12)(13,18,20,15)(14,21,23,24)(16,27,17,25)(19,28,22,26);(5,6)(7,10)(8,9)(11,12)(13,19)(14,16)(15,26)(17,23)(18,28)(20,22)(21,27)(24,25)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=g,e^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, 7533867);