#7678668

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

8252 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)(7,11)(13,14)(15,23)(16,20)(17,18)(19,21)(22,28)(24,25)(26,27);(5,6,9,10)(7,8,11,12)(13,15,19,16)(14,20,21,23)(17,26,22,24)(18,25,28,27);(1,2,3,4)(5,7)(6,8)(9,11)(10,12)(14,21)(15,16)(17,22)(25,27);(1,2)(3,4)(5,8)(6,11)(7,10)(9,12)(13,16,19,15)(14,20,21,23)(17,26,22,24)(18,27,28,25);(13,17,19,22)(14,18,21,28)(15,24,16,26)(20,27,23,25);(13,18)(14,22)(15,25)(16,27)(17,21)(19,28)(20,24)(23,26)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302340, #72302985, #72302991
maximal normal subgroup of	#72302340, #72302985, #72302991

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585864);