#7693682

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

7070 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)(5,6,9,8)(7,11,12,10)(13,14)(15,20)(16,26)(17,21)(18,22)(19,23)(24,25)(27,28);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,15,19,17)(14,20,23,21)(16,27,22,24)(18,25,26,28);(5,7)(6,10)(8,11)(9,12)(13,16,19,22)(14,18,23,26)(15,24,17,27)(20,28,21,25);(5,8,9,6)(7,10,12,11)(13,17,19,15)(14,21,23,20)(16,24,22,27)(18,28,26,25);(13,14)(15,20)(16,18)(17,21)(19,23)(22,26)(24,28)(25,27);(13,18)(14,22)(15,25)(16,23)(17,28)(19,26)(20,24)(21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238246, #72238250, #72238257
maximal normal subgroup of	#72238246, #72238250, #72238257

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7600878);