#7748705

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

8478 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)(5,6)(7,11)(8,9)(10,12)(13,14)(15,20)(16,19)(17,22)(18,21)(23,27)(24,26)(25,28);(1,2)(3,4)(13,15)(14,20)(16,17)(18,23)(19,22)(21,27)(24,25)(26,28);(6,9)(10,11)(14,16)(15,22)(18,26)(25,27);(5,6)(7,10)(8,9)(11,12)(13,16,19,14)(15,17,22,20)(18,24,26,21)(23,25,28,27);(5,7)(6,10)(8,12)(9,11)(13,17)(14,15)(16,22)(18,25)(19,20)(21,23)(24,28)(26,27);(13,18)(14,21)(15,23)(16,24)(17,25)(19,26)(20,27)(22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249491, #72249498, #72249523
maximal normal subgroup of	#72249491, #72249498, #72249523

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7655901);