#7776137

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

8150 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=i,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266742
maximal normal subgroup of	#72266742
maximal quotient of	#72266747, #72266759, #72266767, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7683333);