#7678194

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

7146 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)(5,6,9,7)(8,10,12,11)(13,14)(15,17)(16,23)(18,21)(19,22)(20,24)(25,27)(26,28);(1,2)(3,4)(13,15)(14,21)(16,26)(17,24)(18,20)(19,25)(22,28)(23,27);(6,7)(10,11)(13,16,20,19)(14,22,24,23)(15,25,18,26)(17,28,21,27);(5,7,9,6)(8,11,12,10)(13,17,20,21)(14,15,24,18)(16,27,19,28)(22,26,23,25);(13,18,20,15)(14,21,24,17)(16,25,19,26)(22,28,23,27);(5,8,9,12)(6,10,7,11)(13,19,20,16)(14,23,24,22)(15,25,18,26)(17,28,21,27)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=h,e^2=i,f^2=h,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	#72238189, #72238196, #72238257
maximal normal subgroup of	#72238189, #72238196, #72238257

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585390);