#7626986

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×#525	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6848 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)(15,20)(16,24)(17,22)(18,19)(21,26)(23,25)(27,28);(1,2,4,3)(14,19)(15,22)(21,24)(23,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,22)(14,20,19,17)(16,23,26,28)(21,27,24,25);(7,11)(10,12)(13,16,18,26)(14,21,19,24)(15,23,22,28)(17,25,20,27);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,17,18,20)(14,22,19,15)(16,25,26,27)(21,28,24,23);(5,6)(7,10)(8,9)(11,12)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=g,e^2=gi,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, 7534182);