#7838434

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

7133 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)(15,17)(20,24)(25,27)(26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,24,17,20)(16,21,22,18)(25,26,27,28);(6,8)(10,12)(13,15,19,17)(14,20,23,24)(16,27,22,25)(18,28,21,26);(7,11)(10,12)(13,16,19,22)(14,21,23,18)(15,25,17,27)(20,28,24,26);(5,7)(6,10)(8,12)(9,11)(13,17,19,15)(14,20,23,24)(16,25,22,27)(18,28,21,26);(5,8,9,6)(7,12,11,10)(13,18,19,21)(14,22,23,16)(15,26,17,28)(20,27,24,25)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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	#72308141, #72308392
maximal normal subgroup of	#72308141, #72308392

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7745630);