#7770373

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

5876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,5,6)(3,4,7,8)(9,10,13,15)(11,16,14,12)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(9,11)(10,12)(13,14)(15,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31);(2,6)(3,7)(9,12,13,16)(10,14,15,11)(18,19)(21,29)(23,25)(27,32);(1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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	#72249491, #72249501, #72249532
maximal normal subgroup of	#72249491, #72249501, #72249532

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7677569);