#7670669

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	C₂×#3496	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8582 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)(5,6,9,11)(7,12,8,10)(13,14,19,16)(15,23,24,20)(17,21,27,25)(18,26,28,22);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,23)(17,18)(19,24)(21,22)(25,26)(27,28);(2,4)(5,6)(7,10)(8,12)(9,11)(15,24)(17,27)(20,23)(21,25);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(13,16,19,14)(15,23,24,20)(17,25,27,21)(18,26,28,22);(13,17)(14,21)(15,18)(16,25)(19,27)(20,22)(23,26)(24,28);(13,18)(14,22)(15,17)(16,26)(19,28)(20,21)(23,25)(24,27)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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	#72305253, #72305261, #72305286
maximal normal subgroup of	#72305253, #72305261, #72305286
maximal quotient of	#72306203

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577865);