#7670676

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

7402 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)(13,14)(15,21)(16,19)(17,23)(18,20)(22,27)(24,26)(25,28);(5,6,9,10)(7,11,8,12)(13,15)(14,17)(16,22)(18,23)(19,25)(20,21)(24,27)(26,28);(2,4)(5,7,9,8)(6,11,10,12)(13,14,18,20)(15,21,23,17)(16,24,26,19)(22,25,28,27);(1,2)(3,4)(13,14)(15,17)(16,19)(18,20)(21,23)(22,25)(24,26)(27,28);(5,8,9,7)(6,12,10,11)(13,16,18,26)(14,19,20,24)(15,22,23,28)(17,25,21,27);(13,17)(14,15)(16,25)(18,21)(19,22)(20,23)(24,28)(26,27)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,e^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	#72305286
maximal normal subgroup of	#72305286
maximal quotient of	#72306266

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577872);