#7667013

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72237719, #72240898, #72240901, …
maximal subgroup of	#72236189, #72236637, #72236649, …
maximal normal subgroup of	#72236189, #72236637, #72236649, …
maximal quotient of	C₄×#59816, #72282047, #72297597, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7574209);