#7847206

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

8059 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)(5,6,9,10)(7,8,11,12);(1,2)(3,4)(13,14,18,19)(15,20,25,27)(16,21,22,17)(23,24,26,28);(6,10)(7,11)(14,19)(17,21)(20,27)(24,28);(5,7)(6,8)(9,11)(10,12)(13,15)(14,20)(16,26)(17,24)(18,25)(19,27)(21,28)(22,23);(5,8)(6,7)(9,12)(10,11)(13,16,18,22)(14,21,19,17)(15,23,25,26)(20,24,27,28);(13,17,18,21)(14,22,19,16)(15,24,25,28)(20,23,27,26)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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	#72279637, #72315189, #72315191
maximal normal subgroup of	#72279637, #72315189, #72315191
maximal quotient of	#72280595, #72280903, #72281009

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7754402);