#7688538

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6855 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)(5,6,9,12)(7,10,8,11)(13,14,19,16)(15,23,25,20)(17,21,24,26)(18,22,28,27);(5,6)(7,10)(8,11)(9,12)(13,14,19,16)(15,23,25,20)(17,21,24,26)(18,22,28,27);(2,3)(5,7,9,8)(6,10,12,11)(13,15,19,25)(14,20,16,23)(17,18,24,28)(21,27,26,22);(1,3,4,2)(13,16,19,14)(15,20,25,23)(17,26,24,21)(18,27,28,22);(5,7,9,8)(6,10,12,11)(13,17)(14,21)(15,18)(16,26)(19,24)(20,27)(22,23)(25,28);(5,8,9,7)(6,11,12,10)(13,18)(14,22)(15,24)(16,27)(17,25)(19,28)(20,21)(23,26)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312421, #72315167, #72315192
maximal normal subgroup of	#72312421, #72315167, #72315192

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7595734);