#7693624

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

6923 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)(15,22)(16,19)(17,20)(18,21)(23,27)(24,25)(26,28);(5,6)(7,10)(8,11)(9,12)(13,14)(15,22)(16,19)(17,20)(18,21)(23,27)(24,25)(26,28);(2,3)(5,7,9,8)(6,10,12,11)(13,15,18,20)(14,17,21,22)(16,24,26,23)(19,27,28,25);(1,3,4,2)(13,16,18,26)(14,19,21,28)(15,23,20,24)(17,25,22,27);(5,7,9,8)(6,10,12,11)(13,14)(15,17)(16,19)(18,21)(20,22)(23,25)(24,27)(26,28);(5,8,9,7)(6,11,12,10)(13,17)(14,20)(15,21)(16,25)(18,22)(19,24)(23,28)(26,27)

Polycyclic presentation:

2^9|a^2=ghi,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	#72312837, #72315167, #72315192
maximal normal subgroup of	#72312837, #72315167, #72315192

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7600820);