#7670680

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

7333 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)(3,4)(6,10)(11,12)(14,19)(17,21)(20,26)(23,27);(5,6)(7,11)(8,12)(9,10)(13,14,18,19)(15,20,24,26)(16,21,25,17)(22,27,28,23);(2,4)(14,19)(16,25)(20,26)(22,28);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,15,18,24)(14,20,19,26)(16,22,25,28)(17,23,21,27);(13,16)(14,21)(15,22)(17,19)(18,25)(20,27)(23,26)(24,28);(5,8,9,7)(6,12,10,11)(13,17,18,21)(14,16,19,25)(15,23,24,27)(20,22,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305279, #72305286
maximal normal subgroup of	#72305279, #72305286
maximal quotient of	#72305918, #72306041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577876);