#7670670

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
prime radical	#35358	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7306 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)(6,10)(11,12)(14,20)(18,28)(21,24)(22,26);(5,6)(7,11)(8,12)(9,10)(13,14,19,20)(15,24,16,21)(17,22,27,26)(18,23,28,25);(2,4)(5,7,9,8)(6,11,10,12)(13,15,19,16)(14,21,20,24)(17,25,27,23)(18,22,28,26);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(13,16,19,15)(14,21,20,24)(17,25,27,23)(18,26,28,22);(13,17)(14,22)(15,23)(16,25)(18,21)(19,27)(20,26)(24,28);(13,18)(14,23)(15,22)(16,26)(17,21)(19,28)(20,25)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305286
maximal normal subgroup of	#72305286
maximal quotient of	#72306255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577866);