#7626669

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×#3506	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7254 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)(15,22)(16,25)(17,20)(18,24)(19,21)(23,27)(26,28);(1,3,4,2)(13,15)(14,18)(16,26)(17,23)(19,24)(20,28)(21,22)(25,27);(5,6)(7,10)(8,9)(11,12)(13,16,19,17)(14,20,21,25)(15,23,24,26)(18,27,22,28);(7,11)(10,12)(13,15)(14,18)(16,23)(17,26)(19,24)(20,27)(21,22)(25,28);(5,7)(6,10)(8,11)(9,12)(13,17,19,16)(14,20,21,25)(15,23,24,26)(18,28,22,27);(5,6)(7,10)(8,9)(11,12)(13,18)(14,15)(16,27)(17,28)(19,22)(20,23)(21,24)(25,26)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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, 7533865);