#7689272

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

8422 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:

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

Polycyclic presentation:

2^9|a^2=i,b^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	#72304543, #72304584, #72304609
maximal normal subgroup of	#72304543, #72304584, #72304609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596468);